Quantitative Aptitude

Mensuration

2,507 Questions
All Direct Formula Application 1067 Shape Transformations 343 Mensuration Ratio Equations 277 Combined & Hollow Solids 247 Successive Percentage Changes 178 Inscribed & Circumscribed Figures 165 Cost Estimation Problems 100 Liquid Pour & Submergence 67 Unit Conversions 35 Path & Border Calculations 17 Time & Work Integration 7 Scale & Model Proportions 2

Mensuration Questions

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

If the curved surface area of a right circular cone is $12,320 cm^{2}$ and its base radius is $56$ cm, then its height is $\displaystyle \left(\pi\, =\, \frac{22}{7}\right)$

  1. $42$ cm
  2. $36$ cm
  3. $48$ cm
  4. $50$ cm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Curved surface area of a cone $= \pi rl$, where $r$ is the radius of the cone and $l$ is the slant height.
Hence, curved surface area of this cone, $ = \dfrac {22}{7} \times 56 \times l = 12320 $ 

$ \therefore l = 70  cm $

For a cone, $ l = \sqrt { { h }^{ 2 }+  {r}^{ 2 } } $, where $h$ is the height. 

Hence, $ 70 = \sqrt { { h }^{ 2 }+  {56}^{ 2 } } $ 

$\Rightarrow  4900 = { h }^{ 2 } + 3136 $

$\Rightarrow  { h }^{ 2 } = 1764 $

$ \Rightarrow  h = 42 $ cm

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

A joker's cap is in the form of a right circular cone of base radius $7$ cm and height $24$ cm. Find the area of the sheet required to make $100$ such caps.

  1. $55000{cm}^{2}$
  2. $48724{cm}^{2}$
  3. $30000{cm}^{2}$
  4. $11256{cm}^{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Area of sheet required to make a cap is the Curved surface area of a cone which is $= \pi rl$, where $r$ is the radius of the cone and $l$ is the slant height.

For a cone, $l = \sqrt { { h }^{ 2 }+  {r}^{ 2 } } $, where $h$ is the height. 

Hence, $ l = \sqrt { { 24 }^{ 2 }+  {7}^{ 2 } } $ 

$ \therefore l = 25 $ cm

Hence, area of sheet required to make one cap $

= \dfrac {22}{7} \times 7 \times 25 = 550 $ sq.cm  
Area of sheet required to make $100$ caps $ 100\times 550=55000$ sq.cm

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

The slant height and base diameter of a conical tomb are $25$ m and $14$ m respectively. Find the cost of white washing its curved surface at the rate of Rs. $210$ per $100{m}^{2}$.

  1. Rs. $5627$
  2. Rs. $4156$
  3. Rs. $1155$
  4. Rs. $964$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Curved surface area of a cone $= \pi rl$, where $r$ is the radius of the cone and $l$ is the slant height.
Radius of the tomb $ = \dfrac {\text{Diameter}}{2} = 7  m $
Hence, CSA of this conical tomb , $ = \dfrac {22}{7} \times 7 \times 25 = 550$  sq. m.
Cost of white washing the tomb at Rs. $210$ per $100 {m}^{2} = \dfrac {210}{100} \times 550 =  $ Rs. $  1155 $

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

The radius of a cone is $3\ cm$ and vertical height is $4\ cm$. Find the area of the curved surface.

  1. $62.85\ {cm}^{2}$
  2. $66.05\ {cm}^{2}$
  3. $52.25\ {cm}^{2}$
  4. $47.14\ {cm}^{2}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Curved surface area of a cone $= \pi rl$

For a cone, $l = \sqrt { {h }^{ 2 }+  {r}^{ 2 } } $

Hence, $l = \sqrt { { 3 }^{2 }+  {4}^{ 2 } } $

$ l = 5  cm $
So, CSA of this cone, $ = \cfrac {22}{7} \times 3 \times 5 = 47.14  {cm}^{2} $

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

The radius and height of a cone are in the ratio $4:3$. The area of the base is $154\space cm^2$. Find the area of the curved surface.

  1. $192.5\space cm^2$
  2. $195\space cm^2$
  3. $190.5\space cm^2$
  4. $185.5\space cm^2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let radius be $4x$ and height be $3x$.


$\therefore $ Area of base $=$ $\pi { r }^{ 2 }=154\Rightarrow \dfrac { 22 }{ 7 } \times { \left( 4x \right)  }^{ 2 }=154$


$\Rightarrow \quad 16{ x }^{ 2 }=\dfrac { 154\times 7 }{ 22 } \Rightarrow { x }^{ 2 }=\dfrac { 49 }{ 16 } \Rightarrow x=\dfrac { 7 }{ 4 } $

$\therefore $ Radius $=$ $4\times \dfrac { 7 }{ 4 } =7cm,\quad height=3\times \dfrac { 7 }{ 4 } =\dfrac { 21 }{ 4 } =5.25cm$

Now, CSA of cone $=$ $\pi rl\quad and\quad l=\sqrt { { r }^{ 2 }+{ h }^{ 2 } } =\sqrt { 49+{ \left( 5.25 \right)  }^{ 2 } } $

$\therefore $ CSA $=$ $\dfrac { 22 }{ 7 } \times 7\times 8.75=192.5{ cm }^{ 2 }$

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

The curved surface area of of a right cone is $\displaystyle 286m^{2}$ and its slant height is 13 m then area of the base is

  1. $\displaystyle 286m^{2}$
  2. $\displaystyle 308m^{2}$
  3. $\displaystyle 154m^{2}$
  4. $\displaystyle 187m^{2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Curved surface area of a cone $= \pi rl$  where r is the radius of
the cone and l is the slant height.
Hence, CSA of this cone, $ = \frac {22}{7} \times r \times 13 = 286 $
$ =>r = 7  m $

Area of base $ = \pi {r}^{2} = \frac {22}{7} \times 7 \times 7 = 154  m^2 $

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

The curved surface area of a cone of slant height l and radius r is given by

  1. $ \displaystyle \frac{1}{3}\pi / r^{2} $
  2. $ \displaystyle \pi rl $
  3. $ \displaystyle \pi rl^{2} $
  4. $ \displaystyle \frac{1}{3}\pi rl $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A cone is a three-dimensional geometric shape consisting of all line segments joining a single point to every point of a two-dimensional figure.

Slant height of cone (l)=$\sqrt{r^{2}+h^{2}}$Them covered surface area of cone =$\pi rl$

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

A cylindrical rod of lenght h is meted and cast into a cone of base radius twice that of the cylinder What is the height of the cone?

  1. $\displaystyle \frac{3h}{4}$
  2. $\displaystyle \frac{4h}{4}$

  3. 2h
  4. $\displaystyle \frac{h}{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the radius of  cylindrical rod is r and  cylindrical rod height is h given

Then volume of  cylindrical rod=$\pi r^{2}h$
And volume of cone of base twice the radius of   cylindrical rod=$\frac{1}{3}\pi (2r)^{2}H=\frac{4}{3}\pi r^{2}H$
The cylindrical rod melted and make cone 
$\therefore \frac{4}{3}\pi r^{2}H=\pi r^{2}h\Rightarrow H=\frac{3h}{4}$

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

The canvas required to construct a cone of height $24$ m and base radius $7$ m is

  1. $500$ $ \displaystyle \ \text{m} ^{2} $
  2. $520$ $ \displaystyle \ \text{m} ^{2} $
  3. $550$ $ \displaystyle \ \text{m} ^{2} $

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given the height of cone 24 m and base radius is 24 m

Then slant height =$\sqrt{r^{2}+h^{2}}=\sqrt{(24)^{2}+(7)^{2}}=\sqrt{576+49}=\sqrt{625}=25$
Then canvas required to constrict con=the curved surface area of cone=$\frac{22}{7}\times 7\times 25=550 cm^{2}$

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

Find the surface area of a cone whose radius is $12$ m and the slant height is $23$ m.

  1. $\displaystyle 420\pi { \ mm }^{ 2 }$
  2. $\displaystyle 420\pi {\ cm }^{ 2 }$
  3. $\displaystyle 420\pi \ { m }^{ 2 }$
  4. $\displaystyle 420\pi \ m$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Surface area of cone is $A=πr(r+l)$

Here, radius is $r=12$ m and slant height is $l=23$ cm.

Thus,

$A=πr(r+l)=π\times 12(12+23)=12π\times 35=420π$ m$^2$

Hence, the surface area of the cone is $420π$ m$^2$.

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

Find the surface area of a cone whose radius is $20$ cm and the slant height is $0.3$ cm. (use $\displaystyle \pi =3.14$)

  1. $\displaystyle 1274.84\ m$
  2. $\displaystyle 1274.84{ \ mm }^{ 2 }$
  3. $\displaystyle 1274.84{ \ cm }^{ 2 }$
  4. $\displaystyle 1274.84\ cm$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given, radius $=20$ cm and height $=0.$ cm

Surface area of a cone $=$ $\displaystyle \pi r\left( r+s \right) $
$\displaystyle =3.14\times 20\times \left( 20+0.3 \right) $
$\displaystyle =3.14\times 20\times 20.3$
$\displaystyle =1274.84{ cm }^{ 2 }$

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

The curved surface area of a cone is $\displaystyle 320{\  m }^{ 2 }$ whose radius is $7$ m. Find the surface area of a cone. (Assume $\displaystyle \pi =\frac { 22 }{ 7 } $)

  1. $\displaystyle 474{\ m }^{ 2 }$
  2. $\displaystyle 424{\ m }^{ 2 }$
  3. $\displaystyle 404{ \ m }^{ 2 }$
  4. $\displaystyle 454\ m$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given, surface area of cone $=20$ $m^2$ and radius $7$ m

Curved surface area of a cone $=$ $\displaystyle \pi rs$
$\displaystyle =320+\left( { 22 }/{ 7 } \right) \times 7\times 7$
$\displaystyle =320+154$
$\displaystyle =474{ m }^{ 2 }$

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

Calculate the surface area of a cone whose radius is $\dfrac{1}{3}$ cm and slant height is $12$ cm.

  1. ${3\pi}$ $cm^2$
  2. $\dfrac { 37\pi }{ 9 }$ $ { cm }^{ 2 }$
  3. $\dfrac { 37\pi }{ 8 }$ $ { cm }^{ 2 }$
  4. $\dfrac { 5\pi }{ 9 }$ $ { cm }^{ 2 }$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Surface area of cone is $A=πr(r+l)$


Here, radius is $r=\dfrac { 1 }{ 3 }$ cm and slant height is $l=12$ cm.

Thus,
 
$A=πr(r+l)=π\times \dfrac { 1 }{ 3 } \left( \dfrac { 1 }{ 3 } +12 \right) =\dfrac { 1 }{ 3 } π\times \dfrac { 37 }{ 3 } =\dfrac { 37π }{ 9 }$ cm$^2$
 
Hence, the surface area of the cone is $\dfrac { 37π }{ 9 }$ cm$^2$.

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

A cone has a radius of $2$ cm and height of $3$ cm, find total surface area of the cone.

  1. $\displaystyle 35.168\ cm$
  2. $\displaystyle 36.158{ \ cm }^{ 2 }$
  3. $\displaystyle 35.168{ cm }^{ 2 }$
  4. $\displaystyle 35.168{\ m }^{ 2 }$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

First need to find the value of slant height(s) of a cone, using Pythagoras theorem, since the cross section is a right triangle.
$\displaystyle { s }^{ 2 }={ h }^{ 2 }+{ r }^{ 2 }$
$\displaystyle { s }^{ 2 }={ 3 }^{ 2 }+{ 2 }^{ 2 }$
$\displaystyle { s }^{ 2 }=9+4$
$\displaystyle s=\sqrt { \left( 13 \right)  } $
$\displaystyle s=3.6$ cm
Then, surface area of a cone $\displaystyle =\pi rs+\pi { r }^{ 2 }$
$\displaystyle =3.14\times 2\times 3.6+3.14\times 2\times 2$
$\displaystyle =22.608+12.56$
$\displaystyle= 35.168{ cm }^{ 2 }$

Multiple choice maths solids surface area of a cone surface area of cone the surface area of a cone

A conical water tank has a radius of $0.2$ mm and height of $1.2$ mm, find total surface area of the tank.

  1. $\displaystyle 0.9734\ cm$
  2. $\displaystyle 0.9734{ \ cm }^{ 2 }$
  3. $\displaystyle 0.9734\ mm$
  4. $\displaystyle 0.9734{\ mm }^{ 2 }$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given, radius $=0.2$ mm and height $=1.2$ mm 
Then find the value of slant height(s) of a conical water tank, using Pythagoras theorem, since the cross section is a right triangle.
$\displaystyle { s }^{ 2 }={ h }^{ 2 }+{ r }^{ 2 }$
$\displaystyle { s }^{ 2 }={ 1.2 }^{ 2 }+{ 0.2 }^{ 2 }$
$\displaystyle { s }^{ 2 }=1.44+0.4$
$\displaystyle s=\sqrt { \left( 1.84 \right)  } $
$\displaystyle s=1.35$ mm
So, surface area of a cone $\displaystyle =\pi rs+\pi { r }^{ 2 }$
$\displaystyle =3.14\times 0.2\times 1.35+3.14\times 0.2\times 0.2$
$\displaystyle =0.8478+0.1256$
$\displaystyle =0.9734{\  mm }^{ 2 }$